A 2002 e46 that has not had its cooling system rebuilt is overdue, so it is not surprising that it is loosing water. Could be overflow reservoir cracked, could be thermostat cover. Could be water pump. Could be any of the hoses. Generally, however, you will see some steam coming from somewhere when the car is overheating.
I have two 2004 e46s. Within the past year I have completely replaced both those cooling systems. On the 330, the water pump failed at 100k miles. On the 325, the overflow tank split at 130k miles. On both cars, I replaced everything, radiator, thermostat, water pump, hoses, fan clutch on the 330, everything. I learned on my e36s that I like myself better if I do it all and get it over with all at once.
Filling the overflow tank full but not bleeding the system will leave a lot of air in the system. That will lead to prompt overheating which builds up a lot of pressure and forces what water you have out the leak fairly quickly.
Normally I would not be very concerned about a head gasket, but it sounds like this has been very hot several times, and you mentioned white 'smoke' coming out of the exhaust. The fact that you don't see water in the oil or oil in the water is good, but it is no guarantee that the head gasket has not failed.
Open the bleed screw up near the fill cap, fill the overflow tank clear up with water till you get all the bubbles out of the bleed. Drive the car till it is hot and then shut it off and look under the hood for steam coming out somewhere. If the exhaust still blows steam when the car is fully warmed up, you have big problems.